Incompleteness and incomparability in preference aggregation: complexity results

1
Incompleteness and incomparability in preference
aggregation complexity results
M. Silvia Pini, Francesca Rossi, K. Brent
Venable, and Toby Walsh University of
Padova, Italy NICTA and UNSW, Sydney,
Australia

2
Motivation - (1)

• How to combine preferences of multiple agents in
  presence of incompleteness and incomparability in
  their preference orderings over a set of
  outcomes?
• Incompleteness absence of knowledge on
  relationship between pairs of outcomes
• ongoing preference elicitation
• agents privacy
• Incomparability some elements cannot be compared
• novel incomparable to a biography
• fast expensive car incomparable to slow cheap car

3
Motivation - (2)

• Goal aggregate the agents preferences into a
  single pref. ordering
• Since there are incomplete preferences, we focus
  on computing possible (PW) and necessary winners
  (NW)
• PW outcomes that can be the most preferred ones
  for the agents
• NW outcomes that are always the most preferred
  ones for the agents
• Useful for preference elicitation

4
Outline

• Basic notions on preferences
• Possible and necessary winners
• Computing PW and NW NP-hard
• Approximating PW and NW NP-hard
• Sufficient conditions on preference aggregation
  such that computing PW and NW is polynomial
• How PW and NW are useful in preference elicitation

5
Basic notions - (1)

• Multi-agent scenario each agent expresses his
  preferences via an (incomplete) partial ordering
  over the possible outcomes
• preferences over outcomes A and B
• AgtB or AltB (ordered)
• AB (in a tie)
• AB (incomparable)
• A?B (not specified)
• Example A,B,C outcomes

6
Basic notions - (2)

• Incomplete profile sequence of partial orders
  over outcomes, one for every agent, where at
  least one partial order is incomplete

• Preference aggregation function
  incomplete profiles ? sets of P0s

7
Preference aggregation function example with
Pareto
Pref. aggr. function incomplete profiles ? sets
of P0s
Agent 1
Agent 2
Agent 3
?
gt
gt
A
A
C
C
C
A

gt
gt
gt

gt
B
B
B
only completions that are POs!
8
Possible and necessary winners
Konczac and Lang, 2005

• We extend notions of PW and NW to POs
• Necessary winners
• outcomes which are maximal in every completion
• winners no matter how incompleteness is resolved
• Possible winners
• outcomes which are maximal in at least one of
  the completions
• winners in at least one way in which
  incompleteness is resolved

9
Possible and necessary winners example with
Pareto
Agent 1
Agent 2
Agent 3
?
gt
gt
A
A
C
C
C
A

gt
gt
gt

gt
B
B
B
gt
gt
gt
gt
A
C
A
A
C
A
C
C


gt
gt

gt

gt
B
B
B
B


gt
gt
C
A
A
C
C
A
C
A



gt
gt
gt

gt
B
B
B
B
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PW and NW complexity results

• Computing PW and NW is NP-hard (even
  restricting to incomplete TOs)
• deciding if an outcome is
• a possible winner NP-complete
• a necessary winner coNP-complete
• proof for STV
• Computing good approximations of PW and NW is
  NP-hard
• good approximation for all k, a superset PW
  s.t. PW lt k PW

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Proof idea (for possible winners)

• The proof is for STV (Single Transferrable Vote)
• Reduction from 3-SAT to STV
• Given a 3-SAT formula, we build an STV profile
  such that the possible winners are the
  satisfiable assignments of the 3-SAT problem

12
PW and NW easy from combined result

• Combined result graph where
• nodes candidates
• all arcs
• label of arc A-B set of all relations between A
  and B, such that each relation in at least one
  result
• Given the combined result, PW and NW are easy to
  find
• A in NW if no arc (A-B) with BgtA
• A in PW if all arcs (A-B) with BgtA contain also
  other labels
• Computing the combined result in general NP-hard

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PW and NW a tractable case

• If f is IIA and monotonic
• we can compute an upper approximation (cr) in
  polynomial time
• Also, given cr, polynomial to compute PW and NW
• algorithm not affected by approximation
• IIA when rel(A,B) in the result depends only by
  rel(A,B) given by the agents
• monotonic when we improve an outcome in a
  profile (for ex. we pass from AgtB to AB ), then
  it improves also in the result

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Cr upper approximation of the combined result

• Obtained by
• Considering two profile completions
• (A?B) replaced with (AgtB) for every agent
• (A?B) replaced with (AltB) for every agent
• Then two results (A r1 B) and (A r2 B)
• In cr, put (A r B) where r is r1,r2,everything
  between them
• Order of relations lt, and ?, gt
• f is IIA and monotonic ? cr upper approx.of cr
• Approximation only on arcs with all four labels
• involves only and ?

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cr upper approx.of cr example with Lex
Agent 1
Agent 2
lt
A
A
B
gt
gt
C
PW A,B NW ?
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Computing PW and NW easily

• Input f IIA, monotonic pref. aggr. function,
• ip incomplete profile,
• cr(f,ip) approximation of combined
  result
• Output P, N sets of outcomes
• P??, N??
• foreach A?? do
• if ? C?? s.t. lt ? rel(A,C) then
• N ? N ? A
• if ? C?? s.t. lt rel(A,C) then
• P ? P ? A
• return P,N

It terminates in O(?2) time with NNW and PPW
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IIAmonotone pref. aggr. functions

• Pareto
• Lex
• agents are ordered and, given any two outcomes A
  and B, the relation between them in the result is
  the one given by the first agent in the order
  that doesnt declare AB
• Approval voting
• tractability result already proven in Konczak
  and Lang, 2005 since it is a positional scoring
  rule

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Preference elicitation - (1)

• Process of asking queries to agents in order to
  determine their preferences over outcomes
• 
  Chen and Pu, 2004
• At each stage in eliciting preference there is a
  set of possible and necessary winners
• PW NW ? preference elicitation is over, no
  matter how incompleteness is resolved
• Checking when PW NW hard in general
• 
  Conitzer and Sandholm, 2002
• We prove that pref.elicitation is easy if f IIA
  pol. computable

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Preference elicitation - (2)

• PW NW? preference elicitation is over
• At the beginning NW?
  PW?
• As preferences are declared NW ? PW ?
• If PW ? NW, and A?PW?NW, A can become a loser
  or a necessary winner
• Enough to perform ask(A,B), ?B?PW
• C?PW is a loser ? dominated
• f is IIA ? ask(A.B) involves only A-B
  preferences
• O(PW2) steps to remove incompleteness

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Preference elicitation - (3)

• f is IIA and polynomially computable ?
  determining set of winners via pref. elicitation
  is polynomial in agents and outcomes

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Winner determination

• Input f IIA, pol. computable pref. aggr.
  function,
• P, N set of outcomes
• Output W set of outcomes
• wins bool, P?PW, N?NW
• while P?N do
• choose A?P?N
• wins ? true, Pa ? P ? A
• repeat
• choose B?Pa
• if ?agent s.t. A?B then
• ask(A,B)
• compute f(A,B)
• if f(A,B)(AgtB) then
• P ? P ? B
• if f(A,B)(AltB) then
• P ? P ? A wins ? false
• Pa ? Pa ? B
• until f(A,B) ? (AltB) or Pa ? ?
• if winstrue then

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Main results

• Computing PW and NW NP-hard
• Computing good approximations of PW and NW
  NP-hard
• Computing the combined result NP-hard
• If f IIAmonotonic (and pol. computable) then
• computing an approximation of cr is polynomial
• computing PW and NW is polynomial
• if f IIA pol. computable then
• preference elicitation (i.e., until PWNW) is
  polynomial

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Future work

• adding constraints to agents preferences
• possible and necessary winner must be also
  feasible
• expressing preferences via compact knowledge
  representation formalisms (Ex. CP-nets and soft
  constraints)
• determining PW and NW directly from these compact
  formalisms
• adding possibility distribution over the
  completions of an incomplete preference relation
  between outcomes

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Thank you for your attention!
Questions?
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Incompleteness and incomparability in preference
aggregation complexity results
M. Silvia Pini, Francesca Rossi, K. Brent
Venable, and Toby Walsh University of
Padova, Italy NICTA and UNSW, Sydney,
Australia
Soft06 - Nantes, September 2006
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Motivation-1

• How to combine preferences of multiple agents in
  presence of incompleteness and incomparability in
  their preference orderings over a set of outcomes
• Example
• incompleteness ongoing preference elicitation
• incomparability novel incomparable with a
  biography

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Motivation-2

• Incompleteness absence of knowledge on
  relationship between pairs of outcomes
• ongoing preference elicitation
• agents privacy reasons
• Incomparability we dont want very different
  things to be compared and we may have different
  criteria to optimize
• novel incomparable with a biography
• fast expensive car incomparable with slow cheap
  car

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Basic notions - (3)

• Pareto PO ? PO
• AgtB iff AgtB, for all agents
• AB otherwise

• Preference aggregation function
• incomplete profiles ? sets of P0s

gt
Pareto
C
A


B

C
A
only completions that are POs!


B
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Computing PW and NW - (1)

• Algorithm computing NW and PW in polynomial time,
  given cr
• Input
• f IIA, monotonic pref. aggregation function
• ip incomplete profile over outcomes in ?
• cr(f,ip) approximation of combined result
• Output
• P, N sets of outcomes
• It terminates in O(?2) time with NNW and PPW